Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C26 |
| direct product | C7×#11380 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C7 | C23 |
| commutator | C23 | C22×C7×C8 |
| Frattini, cosocle | C23×C4 | C23×C7 |
| socle, prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C7 ◁ G | |
1246 conjugacy classes of subgroups; only the 406 normal subgroups are shown.
This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 406 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1792 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13)(14,15)(24,25,26,28,27,29,30,31);(9,11)(16,17,19,21)(18,22,23,20);(13,15)(16,18,19,23)(17,20,21,22)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4368);